Flow stability and transition over finite compliant panels 2

Typical applications of compliant coatings according to Gad-el-Hak 1996 include: drag reduction as a result of transition delay, as sound absorbent materials in noisy flow-carrying ducts

Chapter 2 Literature review

Research on compliant coatings could be traced as far back as sixty years ago

This research area had already witnessed an era in the beginning where some earlier researchers were either motivated through the transition delay breakthrough results they obtained or got discouraged along the line due to irreproducibility of some past claimed results, especially through their experimental investigations Typical applications of compliant coatings according

to Gad-el-Hak (1996) include: drag reduction as a result of transition delay, as sound absorbent materials in noisy flow-carrying ducts in aero-engines, and as flexible surfaces to coat naval vessels for the purposes of shielding their sonar arrays from the sound generated by the boundary-layer pressure fluctuations and

of reducing the efficiency of their vibrating metal hulls as sound radiators Compliant coatings had been identified to be cheap and simple method to delay transition Unlike other drag reducing methods such as suction, injection, polymer

or particle additives, passive compliant coatings do not require slots, ducts or internal equipment of any kind

Compliant (membrane) panel is one of the passive ways of applying compliant coatings especially within the boundary layer, and this method had been proved successfully in many theoretical studies in the past, as a possible way of delaying transition farther in a boundary layer flow Investigations into compliant panel research are linked to the way dolphin swims Research work on dolphin fast swimming feature came to limelight with the Gray (1936)’s paradox, where Gray

made an estimate of the power a dolphin could exert based on its physiology, and concluded the power was insufficient to overcome the drag forces in water and finally hypothesized that Dolphin’s skin must have special anti-drag properties After that, studies on the effect of compliant walls on flow stability were inspired

by Kramer (1957)’s observation of swimming dolphins in the late 1950s Kramer assumed that their high propulsive efficiency should be ascribed to the compliance of their skin He then carried out experiments in water by dragging a torpedo covered with a compliant coating conceived to mimic the dolphin’s skin and achieved drag reduction of more than 50 % compared to that of the rigid wall case This achievement of Kramer generated a lot of interests among the researchers after then, which later resulted into various investigations both experimentally and numerically

2.1 Previous experimental investigations on compliant surfaces

After the pioneering achievement of more than 50% drag reduction, Kramer (1960) performed another experiment with streamlined bodies covered with a multi-layer compliant coating whose design and development followed the epidermic structure of the dolphins This second experiment of Kramer (1960) yielded drag reductions of almost 60% Along the line and out of excitements in early 1960s, attempts made to reproduce Kramer’s results by Puryear (1962), Nisewanger (1964), Ritter and Porteous (1965) failed due to experimental difficulties, that is, due to mainly (i) inability to work with flows that are characterized with low turbulence similar to what Kramer used, and (ii) unable to

work with right compliant material properties However, improved experiments were later carried out by Fisher and Blick (1966), Blick and Walters (1968), Chu and Blick (1969) At least their results appeared to yield some positive evidence

of drag reduction, but some of these were later ignored on the basis of experimental irregularities A review of compliant wall drag reduction research up till the mid-1970 was given by Bushnell et al (1977), where they also highlighted the capability of compliant surfaces in turbulent drag reduction

Also, 1970s witnessed a lull in compliant wall research for drag reduction, as many researchers were discouraged to continue with it due to their inability to reproduce Kramer’s claimed results However, 1980s witnessed another form of resurgence to re-evaluate the function of compliant surfaces in boundary layer stabilization and drag reduction, this resurgence was famously led by Carpenter and co-workers Carpenter and Garrad (1985) demonstrated that some selected compliant surfaces are able to delay transition farther and indeed presented an explanation for the failures of the early experiments In a similar experimental work, Gaster (1987) tested samples of compliant layers in a large towing tank at the then National Maritime Institute (UK) The experiments confirmed that viscoelastic compliant layers with the appropriate properties are able to reduce the growth of the TS waves over the corresponding rigid surfaces provided that unstable fluid-surface interaction modes could be avoided or held in check Willis (1986) and Gaster (1987) further demonstrated that some compliant surfaces could reduce the growth of TS waves by an order of magnitude if wall parameters, such as stiffness and damping are optimally tuned or balanced

Lee et al (1995) studied effects of compliant walls on boundary layer instability in a low turbulence wind tunnel and suggested that a delay of transition

in air may be possible As the density of air is much lower than that for the typical compliant materials (about1:800), a large airflow velocity would be needed to generate the perturbation pressure to interact with the compliant walls Carpenter (1998) was consequently critical of their results Unlike the laminar and transitional flows, turbulent flows are much more difficult to be investigated theoretically and experimentally Bushnell et al (1977) considered the effect of compliant walls on turbulent boundary layers and suggested that compliant coatings could possibly modulate the pre-burst flow in a boundary layer through the pressure field In this regard, it was thus plausible that compliant wall could well have yet a favourable influence on skin friction drag in a turbulent boundary layer

Colley et al (1999) carried out experiment to investigate the laminar-to-turbulent transition of the boundary layer over a rotating compliant disc The experiments were carried out under-water using a hot-film probe and the compliant coating was made from vulcanized silicone rubber It was found that compliance had a stabilizing effect on the inviscid instability in the frequency range but an overall destabilizing effect on the boundary-layer flow It was argued that instabilities due to the flexible surface were not prevalent and the overall destabilization was the result of a lowered critical Reynolds number More recent experiments of Colley et al (2006) have verified the results of Cooper and Carpenter (1997) that compliance can destabilize Type-II disturbances on a

rotating compliant disc Their previous experiments were modified to greatly reduce the background noise, allowing the effect of compliance on the Type-II mode to be investigated A new, much softer material was used for the compliant wall

Huang et al (2008) experimentally investigated the effect of compliant surfaces

on the receptivity and bypass transition of a boundary layer Measurements were made in the pre-transitional and transitional boundary layers on nine different compliant and one rigid surfaces with identical geometries Their compliant surfaces were manufactured from gelatine covered by a 10 µm protective PVC film They observed same laminar boundary layer profiles and growth rate results for all the surfaces, but the receptivity of the laminar boundary layer to free stream disturbances increased close to the leading edge of each compliant surface They recorded the transition onset position on the compliant surface to range from 3% downstream to 20% upstream of the rigid surface position

Erdmann et al (2011) used different types of voice-coil- and piezo-driven membrane actuators effectively to introduce counter waves into the boundary layer to cancel the travelling TS waves They were able to shift the transition region to about seven TS wavelengths (≈ 45 mm) Recently, Patzold et al (2013) used actively driven compliant wall which was integrated as part of wing’s surface to delay transition through attenuation of convective instabilities With this approach, piezo-polymer actuation elements in combination with model predictive control algorithms attenuated the local TS-wave amplitudes by 85.4%,

and transition location also shifted towards the wing’s trailing edge by ∆x ≈ 100

mm

2.2 Previous numerical or theoretical investigations on compliant surfaces

Various numerical methods had been developed since almost three decades ago for investigating boundary-layer instability, transition and transition control especially for over the rigid walls, but which could as well be extended to over compliant surface investigations These methods include: Turbulence Modeling (TM) (Zhang et al (1998)), Large Eddy Simulation (LES) (Ducros et al (1996)), Linear Stability Theory (LST) (Reed et al (1996)), Parabolized Stability Equation (PSE) (Herbert (1997)) and Direct Numerical Simulation (DNS) (Kleiser and Zang (1991)) Since this present work concerns DNS, the review will first focus

on DNS carried out on boundary layer stability over compliant surfaces and to later mention other previous numerical works

2.2.1 Boundary layer compliant surface simulations based on DNS approach

With the computers becoming more powerful in terms of memory and processing speeds these days, using direct numerical simulation (DNS) approach for flow stability and transition problems especially within the boundary layer is

no more a thing of concern Irrespective of whether all the nonlinear terms are included in the flow governing equations or not, DNS provides the most accurate way to investigate both unstable and transitional flows From the literature, two types of DNS method that have been used in the direct simulation of boundary

layer transition problems include: (i) Spatial DNS (SDNS) and (ii) Temporal DNS (TDNS)

For the SDNS, the actual boundary layer is taken into considerations in the computation of the evolving perturbations More peculiarity about this SDNS approach is that, it always leads to more complex implementation of both the inflow and outflow boundary conditions Since for the past decade up till now, more researchers still prefer the SDNS approach over the TDNS counterpart, because SDNS can guarantee the real simulation of experiments On the other hand for TDNS and in order to make the computation simpler, the perturbed flow

is assumed to be periodic in the streamwise direction The purpose of this is to allow the streamwise length of the computational domain to be truncated to only one or just a few primary instability wavelengths The main noted drawback with TDNS approach is the discrepancy between the computational model and the physical flow itself Spalart and Yang (1987) used this approach in their simulations despite the drawback

Probably the first temporal direct numerical simulation of boundary-layer waves over a compliant surface (tensioned membrane to be specific) was performed by Domaradzki and Metacalfe (1987) In their simulation, the Fourier series and Chebyshev expansion method are employed in the streamwise and normal directions respectively For simplicity linearized boundary conditions are applied They studied the temporal and spatial behaviour of the terms in the kinetic energy balance equation and verified the class A and class B character of the computed waves Hall (1988) also developed a temporal simulation algorithm

for simulating 2D instability waves over soft polyvinyl chloride (PVC) layers This work seems to be the only computational study on volume-based walls The transient finite element DYNA2D code was utilized to model the solid wall equations Explicit iteration procedure was adopted for the fluid and solid coupling Three materials including a soft polyvinyl chloride (PVC), stiffer PVC and a two-layer material consisting of a thick layer of soft PVC covered by a thin layer of neoprene were investigated Though much attention was given to modelling solid deformation, little information about flow field was provided by Hall (1988)

Metcalfe et al (1991) reported their 3D temporal DNS work on boundary layer flow instability over a compliant panel Their simulation showed that nonlinear secondary instabilities could arise and cause the flow to become unstable when it was predicted to be stable by linear theory Therefore, besides linear stability analysis, the nonlinear mechanisms also require much attention for optimizing compliant surfaces to delay transition They also found that nonlinear interactions among the different classes of compliant wall modes appear to require commensurate phase speeds and the phase relationship between the modes can strongly affect their interaction Λ-vortices that are similar to the flow structures

in boundary-layer transition over a rigid wall were also observed Furthermore, they found that by carefully choosing compliant wall parameters, they could inhibit the formation of a strong spanwise vorticity spike in the detached shear layer above the wall Although their work was based on temporal theory and linear boundary condition assumption, it indicated that the task of studying and

optimizing 3D compliant surfaces to reduce drag or delay transition is worth pursuing

Davies and Carpenter (1997)’s simulation of boundary-layer stability over finite compliant panel was perhaps the first work done on the linear Navier-Stokes simulations of flow stability over a compliant surface A novel vorticity-velocity method was used in the simulations Using this method with special treatment for the wall and fluid inertial terms, they solved the linearized N-S equations and presented the results for the spatial evolution of TS waves The complex response

of finite panels was investigated in great detail By choosing the frequency of the

TS wave above the cut-off frequency of the compliant panel, Davies and Carpenter showed that the response of the panel actually consists of superposition

of three eigenmodes – one original TS mode and two FISI (or CIFI) modes Moreover, their results displayed the complicated wave effects introduced by the edge of a finite panel; in particular, the interaction between the TS waves and the leading edge of the compliant panel The main conclusion drawn by Davies and Carpenter from their simulations is that panels as short as one TS wavelength remain effective at suppressing TS waves They also demonstrated that certain very short compliant panels are even more effective at wave suppression than longer ones with the same properties

Wiplier and Ehrenstein (2000, 2001) adopted the primitive-variable method to simulate the spatial evolution of 2D disturbances in a boundary-layer flow over compliant membranes and panels The behaviour of the disturbances as convective and absolute instabilities was investigated Their simulation results

re-affirmed the validity of the linear stability theory and show that absolute instability could arise from the coalescence between an upstream propagating evanescent mode and downstream propagating TS wave, as was suggested by Yeo

et al (1996) Their model takes into account the non-parallelism of the flow and nonlinear effects within the flow To handle the moving boundary problem, the physical domain was transformed into a fixed computational domain However, as the prescribed base flow was dynamically stretched with the deformation of the compliant boundary, this may well introduce an unknown error into the results It

is hard to be convinced that such an approach will work for other than small amplitude surface waves; in which case the condition at the boundary will be quite similar to that modelled by linear flow-wall interaction conditions When the amplitude of the surface waves becomes nonlinearly significant, the stretching of the base flow may well introduce artificial dynamics that have to be accounted for

Wang et al (2001) employed a 2D vorticity-streamfunction method for spatially simulating the unsteady waves over finite-length membranes Two cases with different tensions were investigated in some details The results were compared with those for a rigid wall Davies and Carpenter (2001) developed a method for simulating linear disturbance evolution in 3D boundary layer over compliant surfaces and applied it to boundary-layer flow on a rotating disk Unlike normal 3D vorticity-velocity method in which six governing equations are usually required, only three governing equations are solved in their highly efficient method The linearized form of this method was validated for the case of